Calculus Examples

$lim_{t \to - 2} {(t + 1)}^{9} (t^{2} - 1)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $- 2$ .

$lim_{t \to - 2} {(t + 1)}^{9} \cdot lim_{t \to - 2} t^{2} - 1$

Step 2

Move the exponent $9$ from ${(t + 1)}^{9}$ outside the limit using the Limits Power Rule.

${(lim_{t \to - 2} t + 1)}^{9} \cdot lim_{t \to - 2} t^{2} - 1$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $- 2$ .

${(lim_{t \to - 2} t + lim_{t \to - 2} 1)}^{9} \cdot lim_{t \to - 2} t^{2} - 1$

Step 4

Evaluate the limit of $1$ which is constant as $t$ approaches $- 2$ .

${(lim_{t \to - 2} t + 1)}^{9} \cdot lim_{t \to - 2} t^{2} - 1$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $- 2$ .

${(lim_{t \to - 2} t + 1)}^{9} \cdot (lim_{t \to - 2} t^{2} - lim_{t \to - 2} 1)$

Step 6

Move the exponent $2$ from $t^{2}$ outside the limit using the Limits Power Rule.

${(lim_{t \to - 2} t + 1)}^{9} \cdot ({(lim_{t \to - 2} t)}^{2} - lim_{t \to - 2} 1)$

Step 7

Evaluate the limit of $1$ which is constant as $t$ approaches $- 2$ .

${(lim_{t \to - 2} t + 1)}^{9} \cdot ({(lim_{t \to - 2} t)}^{2} - 1 \cdot 1)$

Step 8

Evaluate the limits by plugging in $- 2$ for all occurrences of $t$ .

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Step 8.1

Evaluate the limit of $t$ by plugging in $- 2$ for $t$ .

${(- 2 + 1)}^{9} \cdot ({(lim_{t \to - 2} t)}^{2} - 1 \cdot 1)$

Step 8.2

Evaluate the limit of $t$ by plugging in $- 2$ for $t$ .

${(- 2 + 1)}^{9} \cdot ({(- 2)}^{2} - 1 \cdot 1)$

Step 9

Simplify the answer.

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Step 9.1

Add $- 2$ and $1$ .

${(- 1)}^{9} \cdot ({(- 2)}^{2} - 1 \cdot 1)$

Step 9.2

Raise $- 1$ to the power of $9$ .

$- 1 \cdot ({(- 2)}^{2} - 1 \cdot 1)$

Step 9.3

Simplify each term.

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Step 9.3.1

Raise $- 2$ to the power of $2$ .

$- 1 \cdot (4 - 1 \cdot 1)$

Step 9.3.2

Multiply $- 1$ by $1$ .

$- 1 \cdot (4 - 1)$

Step 9.4

Subtract $1$ from $4$ .

$- 1 \cdot 3$

Step 9.5

Multiply $- 1$ by $3$ .

$- 3$